3.1377 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=106 \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(2943 + 4097*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)^2) + ((54 + 491*x)*(2 + 3*x
^2)^(3/2))/(840*(3 + 2*x)^4) - (3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (39663*ArcT
anh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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Rubi [A]  time = 0.181528, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^5,x]

[Out]

(3*(2943 + 4097*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)^2) + ((54 + 491*x)*(2 + 3*x
^2)^(3/2))/(840*(3 + 2*x)^4) - (3*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (39663*ArcT
anh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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Rubi in Sympy [A]  time = 18.6574, size = 97, normalized size = 0.92 \[ - \frac{3 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{32} - \frac{39663 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1372000} + \frac{\left (196656 x + 141264\right ) \sqrt{3 x^{2} + 2}}{313600 \left (2 x + 3\right )^{2}} + \frac{\left (982 x + 108\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{1680 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**5,x)

[Out]

-3*sqrt(3)*asinh(sqrt(6)*x/2)/32 - 39663*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*
sqrt(3*x**2 + 2)))/1372000 + (196656*x + 141264)*sqrt(3*x**2 + 2)/(313600*(2*x +
3)**2) + (982*x + 108)*(3*x**2 + 2)**(3/2)/(1680*(2*x + 3)**4)

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Mathematica [A]  time = 0.178605, size = 102, normalized size = 0.96 \[ \frac{-118989 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+\frac{70 \sqrt{3 x^2+2} \left (250602 x^3+559764 x^2+718441 x+245943\right )}{(2 x+3)^4}+118989 \sqrt{35} \log (2 x+3)-385875 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{4116000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^5,x]

[Out]

((70*Sqrt[2 + 3*x^2]*(245943 + 718441*x + 559764*x^2 + 250602*x^3))/(3 + 2*x)^4
- 385875*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 118989*Sqrt[35]*Log[3 + 2*x] - 118989*Sq
rt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/4116000

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Maple [B]  time = 0.018, size = 194, normalized size = 1.8 \[ -{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{211}{117600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{999}{686000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{5779}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{13221}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{7227\,x}{686000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{39663}{1372000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{39663\,\sqrt{35}}{1372000}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{17337\,x}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^5,x)

[Out]

-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)-211/117600/(x+3/2)^3*(3*(x+3/2)^
2-9*x-19/4)^(5/2)-999/686000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-5779/1200500
0/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+13221/6002500*(3*(x+3/2)^2-9*x-19/4)^(3/2
)-7227/686000*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+39663/1372000*(12*(x+3/2)^2-36*x-19
)^(1/2)-39663/1372000*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-
19)^(1/2))+17337/12005000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-3/32*arcsinh(1/2*x*6^(1
/2))*3^(1/2)

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Maxima [A]  time = 0.782618, size = 247, normalized size = 2.33 \[ \frac{2997}{686000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{211 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{14700 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{999 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{171500 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{7227}{686000} \, \sqrt{3 \, x^{2} + 2} x - \frac{3}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{39663}{1372000} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{39663}{686000} \, \sqrt{3 \, x^{2} + 2} - \frac{5779 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{686000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^5,x, algorithm="maxima")

[Out]

2997/686000*(3*x^2 + 2)^(3/2) - 13/140*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*
x^2 + 216*x + 81) - 211/14700*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9
99/171500*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 7227/686000*sqrt(3*x^2 + 2)*x -
3/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 39663/1372000*sqrt(35)*arcsinh(3/2*sqrt(6
)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 39663/686000*sqrt(3*x^2 + 2) - 57
79/686000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.304995, size = 236, normalized size = 2.23 \[ \frac{\sqrt{35}{\left (11025 \, \sqrt{35} \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 4 \, \sqrt{35}{\left (250602 \, x^{3} + 559764 \, x^{2} + 718441 \, x + 245943\right )} \sqrt{3 \, x^{2} + 2} + 118989 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{8232000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^5,x, algorithm="fricas")

[Out]

1/8232000*sqrt(35)*(11025*sqrt(35)*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x +
81)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 4*sqrt(35)*(250602*x^3 + 559764
*x^2 + 718441*x + 245943)*sqrt(3*x^2 + 2) + 118989*(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/
(4*x^2 + 12*x + 9)))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.348378, size = 333, normalized size = 3.14 \[ -\frac{39663}{1372000} \, \sqrt{35}{\rm ln}\left (-9 \, \sqrt{35} + 35 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{35 \, \sqrt{35}}{2 \, x + 3}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + \frac{3}{32} \, \sqrt{3}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{470400} \,{\left (\frac{35 \,{\left (\frac{35 \,{\left (\frac{1365 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 1193 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 16227 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 125301 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^5,x, algorithm="giac")

[Out]

-39663/1372000*sqrt(35)*ln(-9*sqrt(35) + 35*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2
+ 3) + 35*sqrt(35)/(2*x + 3))*sign(1/(2*x + 3)) + 3/32*sqrt(3)*ln(1/2*abs(-2*sqr
t(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(sqrt(
3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sign(1/(2*x
+ 3)) - 1/470400*(35*(35*(1365*sign(1/(2*x + 3))/(2*x + 3) - 1193*sign(1/(2*x +
3)))/(2*x + 3) + 16227*sign(1/(2*x + 3)))/(2*x + 3) - 125301*sign(1/(2*x + 3)))
*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)